2021/12/01 by Delio Mugnolo, Mugnolo, Delio, Marvin Plümer +1
Materials Science · Mathematics · Physics and Astronomy · #34B45 (05C50 35P15 81Q35) #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #Graphene research and applications #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2112.00782
openalex publication_date 2021/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the theory of torsional rigidity -- a quantity routinely considered for Dirichlet Laplacians on bounded planar domains -- for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin Inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph's vertices and is, thus, much easier than computing eigenvalues.